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Type ${\text{III}}_{0}$ cocycles without unbounded gaps

Commentationes Mathematicae Universitatis Carolinae

An example of type III${}_{0}$ cocycle without unbounded gaps of an ergodic probability measure preserving transformation will be shown.

On measure theoretical analogues of the Takesaki structure theorem for type III factors

Colloquium Mathematicae

The orbit equivalence of type $II{I}_{0}$ ergodic equivalence relations is considered. We show that it is equivalent to the outer conjugacy problem for the natural trace-scaling action of a countable dense ℝ-subgroup by automorphisms of the Radon-Nikodym skew product extensions of these relations. A similar result holds for the weak equivalence of arbitrary type $II{I}_{0}$ cocycles with values in Abelian groups.

Mod 2 normal numbers and skew products

Studia Mathematica

Let E be an interval in the unit interval [0,1). For each x ∈ [0,1) define dₙ(x) ∈ 0,1 by $dₙ\left(x\right):={\sum }_{i=1}^{n}{1}_{E}\left({2}^{i-1}x\right)\left(mod2\right)$, where t is the fractional part of t. Then x is called a normal number mod 2 with respect to E if ${N}^{-1}{\sum }_{n=1}^{N}dₙ\left(x\right)$ converges to 1/2. It is shown that for any interval E ≠(1/6, 5/6) a.e. x is a normal number mod 2 with respect to E. For E = (1/6, 5/6) it is proved that ${N}^{-1}{\sum }_{n=1}^{N}dₙ\left(x\right)$ converges a.e. and the limit equals 1/3 or 2/3 depending on x.

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